Prove that metric space is complete/incomplete

Question

I have a function $\rho(x,y) = |x^{1/3} - y^{1/3}|$ and I need to prove if the function is metric, and if it is, the next step is to prove if this metric space is complete.

So metric is a function with following conditions:

1. $\rho(x,y) \iff y=x$
2. $\rho(x,y) = \rho(y,x)$ because of absolute value's properties
3. $\rho(x,z) \leq \rho(x,y) + \rho(y,z)$ — I can't prove it myself

But I know that the function is metric indeed and how do I prove that this metric space is complete?

Answer 1

I will assume that your space is $\mathbb R$. Let $d$ be the usual metric on $\mathbb R$. Then$$\begin{array}{ccc}(\mathbb R,\rho)&\longrightarrow&(\mathbb R,d)\\x&\mapsto&\sqrt[3]x\end{array}$$is an isometry. So, since $(\mathbb R,d)$ is complete, so is your space.